Theorem univ 4473

Description: The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)

Assertion

Ref	Expression
univ	⊢ ∪ V = V

Proof of Theorem univ

Step	Hyp	Ref	Expression
1		pwv 3806	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 3817	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 4214	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2200	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1353  Vcvv 2737  𝒫 cpw 3574  ∪ cuni 3807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-uni 3808

This theorem is referenced by: (None)